基于9 阶van der Pol 方程的三稳态电路设计及实现
Design and Implementation of Tri-Stable Circuit Based on the Ninth-Order van der Pol Equation

基于9 阶van der Pol 方程的分岔结果，设计了1 个平衡点和2 个极限环共存的三稳态电路．利用平均法分 析了9 阶van der Pol 方程的分岔性质，设计了能够实现三稳态现象的无量纲方程的系统参数．根据基尔霍夫电路定 理，利用运算放大器和模拟乘法器等元件，构建了9 阶van der Pol 方程的电路原理图，并通过PSpice仿真和硬件电 路试验验证了该电路的可行性和可靠性．试验结果表明，该电路系统中有1 个稳定平衡点与2 个稳定极限环共存的 现象，为研究确定性激励以及随机激励下三稳态系统的动力学行为奠定了基础．
A kind of tri-stable circuit with one stable equilibrium point and two stable limit cycles was designed based on the bifurcation analysis of the ninth-order van der Pol equation．By using the averaging method，the bifurcation properties of the ninth-order van der Pol equation were analyzed，and the dimensionless equation parameters which could achieve the tri-stable phenomena were designed．According to Kirchhoff laws，the circuit diagram of the ninthorder van der Pol equation was constructed with such elements as operational amplifiers，analog multiplier，etc．The feasibility and reliability of the circuit were verified by PSpice simulation and hardware experiments．The results show that the proposed circuit system does have one stable equilibrium point and two stable limit cycles，which lays a foundation for further study on the dynamic behaviors of tri-stable system under deterministic excitation and random excitation